Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization

We introduce a new variational method for the numerical homogenization of divergence form elliptic, parabolic and hyperbolic equations with arbitrary rough ($L^\infty$) coefficients. Our method does not rely on concepts of ergodicity or scale-separation but on compactness properties of the solution space and a new variational approach to homogenization. The approximation space is generated by an interpolation basis (over scattered points forming a mesh of resolution $H$) minimizing the $L^2$ norm of the source terms; its (pre-)computation involves minimizing $\mathcal{O}(H^{-d})$ quadratic (cell) problems on (super-)localized sub-domains of size $\mathcal{O}(H \ln (1/ H))$. The resulting localized linear systems remain sparse and banded. The resulting interpolation basis functions are biharmonic for $d\leq 3$, and polyharmonic for $d\geq 4$, for the operator -\diiv(a\nabla \cdot) and can be seen as a generalization of polyharmonic splines to differential operators with arbitrary rough coefficients. The accuracy of the method ($\mathcal{O}(H)$ in energy norm and independent from aspect ratios of the mesh formed by the scattered points) is established via the introduction of a new class of higher-order Poincar\'{e} inequalities. The method bypasses (pre-)computations on the full domain and naturally generalizes to time dependent problems, it also provides a natural solution to the inverse problem of recovering the solution of a divergence form elliptic equation from a finite number of point measurements.Comment: ESAIM: Mathematical Modelling and Numerical Analysis. Special issue (2013

Bayesian Numerical Homogenization

Numerical homogenization, i.e. the finite-dimensional approximation of solution spaces of PDEs with arbitrary rough coefficients, requires the identification of accurate basis elements. These basis elements are oftentimes found after a laborious process of scientific investigation and plain guesswork. Can this identification problem be facilitated? Is there a general recipe/decision framework for guiding the design of basis elements? We suggest that the answer to the above questions could be positive based on the reformulation of numerical homogenization as a Bayesian Inference problem in which a given PDE with rough coefficients (or multi-scale operator) is excited with noise (random right hand side/source term) and one tries to estimate the value of the solution at a given point based on a finite number of observations. We apply this reformulation to the identification of bases for the numerical homogenization of arbitrary integro-differential equations and show that these bases have optimal recovery properties. In particular we show how Rough Polyharmonic Splines can be re-discovered as the optimal solution of a Gaussian filtering problem.Comment: 22 pages. To appear in SIAM Multiscale Modeling and Simulatio

Modeling and simulation in tribology across scales: An overview

This review summarizes recent advances in the area of tribology based on the outcome of a Lorentz Center workshop surveying various physical, chemical and mechanical phenomena across scales. Among the main themes discussed were those of rough surface representations, the breakdown of continuum theories at the nano- and micro-scales, as well as multiscale and multiphysics aspects for analytical and computational models relevant to applications spanning a variety of sectors, from automotive to biotribology and nanotechnology. Significant effort is still required to account for complementary nonlinear effects of plasticity, adhesion, friction, wear, lubrication and surface chemistry in tribological models. For each topic, we propose some research directions

Advances in multiscale numerical and experimental approaches for multiphysics problems in porous media

Research on the scientific and engineering problems of porous media has drawn increasing attention in recent years. Digital core analysis technology has been rapidly developed in many fields, such as hydrocarbon exploration and development, hydrology, medicine, materials and subsurface geofluids. In summary, science and engineering research in porous media is a complex problem involving multiple fields. In order to encourage communication and collaboration in porous media research using digital core technology in different industries, the 5th International Conference on Digital Core Analysis & the Workshop on Multiscale Numerical and Experimental Approaches for Multiphysics Problems in Porous Media was held in Qingdao from April 18 to 20, 2021. The workshop was jointly organized by the China InterPore Chapter, the Research Center of Multiphase Flow in Porous Media at the China University of Petroleum (East China) and the University of Aberdeen with financial support from the National Sciences Foundation of China and the British Council. Due to the current pandemic, a hybrid meeting was held (participants in China met in Qingdao, while other participants joined the meeting online), attracting more than 150 participants from around the world, and the latest multi-scale simulation and experimental methods to study multi-field coupling problems in complex porous media were presented.Cited as: Yang, Y., Zhou, Y., Blunt, M. J., Yao, J., Cai, J. Advances in multiscale numerical and experimental approaches for multiphysics problems in porous media. Advances in Geo-Energy Research, 2021, 5(3): 233-238, doi: 10.46690/ager.2021.03.0

SOLID-SHELL FINITE ELEMENT MODELS FOR EXPLICIT SIMULATIONS OF CRACK PROPAGATION IN THIN STRUCTURES

Crack propagation in thin shell structures due to cutting is conveniently simulated using explicit finite element approaches, in view of the high nonlinearity of the problem. Solidshell elements are usually preferred for the discretization in the presence of complex material behavior and degradation phenomena such as delamination, since they allow for a correct representation of the thickness geometry. However, in solid-shell elements the small thickness leads to a very high maximum eigenfrequency, which imply very small stable time-steps. A new selective mass scaling technique is proposed to increase the time-step size without affecting accuracy. New ”directional” cohesive interface elements are used in conjunction with selective mass scaling to account for the interaction with a sharp blade in cutting processes of thin ductile shells

Interplay of Theory and Numerics for Deterministic and Stochastic Homogenization

The workshop has brought together experts in the broad field of partial differential equations with highly heterogeneous coefficients. Analysts and computational and applied mathematicians have shared results and ideas on a topic of considerable interest both from the theoretical and applied viewpoints. A characteristic feature of the workshop has been to encourage discussions on the theoretical as well as numerical challenges in the field, both from the point of view of deterministic as well as stochastic modeling of the heterogeneities